class: left, middle, inverse, title-slide # Investment without Coordination Failures
### Brian C. Albrecht
University of Minnesota
Department of Economics
### January 31, 2020
Paper
bit.ly/Albrecht-JMP
Slides
bit.ly/Albrecht-JMP-Slides
--- <img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_D.png" class="center60"> -- - Two pure-strategy, Nash equilibria 1. Pareto optimal, efficient investment: (Invest, Invest) 2. Pareto dominated, **coordination failure**: (Don't Invest, Don't Invest) -- - Both pure-strategy are stable; mixed-strategy is not --- Competitive Coordination ==================================== - What if there are **many players** and they are **operating in a market**? -- - These features matter because important real-world investments are sunk before entering a market with many players -- - Invest in education before being hired - Produce goods before having a buyer <!--- - Write JMP before knowing schools want a general equilibrium theorist ---> -- - When investments are sunk, *i.e.* when markets are incomplete, coordination failures can arise even with competitive markets - Makowski & Ostroy (1995), Cole, Malaith, & Postlewaite (1999a, 1999b), Makowski (2004), Nöldeke & Samuelson (2014), Felli & Roberts (2016) -- - Previous literature only studied **existence of coordination failures** --- My Question ==================================== - Are the market investments we observe likely to be inefficient? - **Are coordination failures robust**? - Should we predict coordination failures? -- ## My Answer - **No**, if markets are competitive - Formally, inefficient equilibria are not robust to errors in a trembling-hand refinement `\(\Rightarrow\)` We should predict efficient market outcomes --- Mechanism ==================================== - In competitive markets, players only care about **prices** -- - To not invest, each entreprenuer must conjecture she would get paid less than 1 if she invests -- - **Somebody must be wrong** -- - Main Results: - Prop 1: Conjectures must contradict during any coordination failure -- - Theorem 1: Contradictory conjectures are not robust to trembles - `\(\Rightarrow\)` Coordination failures are not robust --- Related Literature ==================================== <font color=#125972><b>Non-cooperative before market/cooperative game:</b></font> Makowski & Ostroy (1995), Cole, Malaith, & Postlewaite (1999), Makowski (2004), Brandenburger & Stuart (2007), Nöldeke & Samuelson (2014) - Contribution: **focus on robust equilibria** via a refinement --- Related Literature ==================================== <font color=#125972><b>Non-cooperative before market/cooperative game:</b></font> Makowski & Ostroy (1995), Cole, Malaith, & Postlewaite (1999), Makowski (2004), Brandenburger & Stuart (2007), Nöldeke & Samuelson (2014) - Contribution: focus on robust equilibria via a refinement <font color=#125972><b>Competitive equilibrium refinements:</b></font> Gale (1992, 1996), Dubey & Geanakoplos (2002), Dubey, Geanakoplos, & Shubik (2004), Zame (2007), Scheuer & Smetters (2018) - Contribution: study **dynamic models with coordination failures** --- Related Literature ==================================== <font color=#125972><b>Non-cooperative before market/cooperative game:</b></font> Makowski & Ostroy (1995), Cole, Malaith, & Postlewaite (1999), Makowski (2004), Brandenburger & Stuart (2007), Nöldeke & Samuelson (2014) - Contribution: focus on robust equilibria via a refinement <font color=#125972><b>Competitive equilibrium refinements:</b></font> Gale (1992, 1996), Dubey & Geanakoplos (2002), Dubey, Geanakoplos, & Shubik (2004), Zame (2007), Scheuer & Smetters (2018) - Contribution: study dynamic models with coordination failures <font color=#125972><b>Coordination failures are not robust:</b></font> Albrecht (2016), Penta & Zuazo-Garin (2018) - Contribution: **Competition rules out** coordination failures --- Road Map for Talk ==================================== 1. **Example** - Environment - Equilibrium - Refinement 2. General Matching Model - Competitive Equilibria with Fixed Investments - Ex Post Efficiency 3. Investment Game where Players Choose Investments - Investment Equilibria - Perfect Investment Equilibria - Ex Ante Efficiency --- Sketch of General Environment ==================================== - Continuum of buyers and sellers, finite types - Transferable utility -- .center[] ??? Simple set up to focus on the coordination aspect Ignore hold-up, bargaining issues --- Example ==================================== - Agents are endowed with a type `\(t \in T = \{\beta, \sigma\}\)`: "buyer" or "seller" - Measure one of both - Buyers choose an investment `\(b \in B= \{0,1\}\)` - Sellers choose an investment `\(s \in S = \{0,1\}\)` -- - Cost to buyer = `\(\frac{1}{4}b\)` - Cost to seller = `\(\frac{1}{4}s\)` - Surplus `\(v(b,s) = bs\)` of matching --- Stage 2: Matching Market =================================== - A buyer `\(b\)` chooses a **match contract** `\((b,s) \in \left\{(b,0) , (b,1)\right\}\)` guaranteeing the right to join a match with `\(s\)` - A seller `\(s\)` chooses `\((b,s) \in \left\{(0,s) , (1,s)\right\}\)` -- - Matches have competitive prices: `\(p : B \times S \to \mathbb{R}\)` - `\(p(b,s)\)`: transfer from buyer with `\(b\)` to seller with `\(s\)` -- - Matching market indirect utility for `\(b\)`: `$$v^*_b(p) = \max_{s}\left\{ v(b,s) - p(b,s) \right\}$$` -- - Matching market indirect utility for `\(s\)`: `$$v^*_s(p) = \max_{b}\left\{ p(b,s) \right\}$$` --- Stage 1: Decision Problem ============================== - Agents only have **price conjectures**: `\(\tilde{p}^t : B \times S \to \mathbb{R} \quad \forall t \in \{\beta,\sigma\}\)` - Not a distribution but can think of conjectures = expected price -- - Buyer's investment problem: `$$\max_{b} \left\{\underbrace{ v^*_b\left(\tilde{p}^\beta\right)}_{\text{Utility from Conjectured Optimal Contract}} - \underbrace{\frac{1}{4}b}_{\text{Investment Cost}}\right\}$$` -- - Seller's investment problem: `\(\max_{s} \left\{ v^*_s\left(\tilde{p}^\sigma\right) - \frac{1}{4}s\right\}\)` --- Investment Equilibrium ==================================== Equilibrium is a vector of matches, prices, and conjectures, such that -- 1. Each buyer chooses `\((b,s)\)` to maximize utility, given `\(\tilde{p}^\beta(b,s)\)` 2. Each seller chooses `\((b,s)\)` to maximize utility, given `\(\tilde{p}^\sigma(b,s)\)` -- 3. Prices clear matching markets - Mass of buyers choosing `\((b,s)\)` = mass of sellers choosing `\((b,s)\)` -- 4. Rational conjectures: conjectures are not contradicted by the data - If positive mass of agents choose contract `\((b,s)\)`, conjectures agree with the posted price: `\(\tilde{p}^\beta (b,s)=\tilde{p}^\sigma (b,s)= p(b,s)\)` - If zero mass choose `\((b,s)\)`, conjectures are not pinned down ??? Informal Definition --- Payoffs ===================================== .center[] -- - Prices and conjectures are equilibrium objects - Rational conjectures: if a match `\((b,s)\)` is chosen, `\(\tilde{p}^\beta (b,s)= \tilde{p}^\sigma (b,s)= p(b,s)\)` - Allows disagreement for matches that do not occur ??? We need to find conjectures and prices --- Constructing Equilibria ==================================== - Efficient equilibrium: everyone invests - Suppose realized prices: `\(p(1,1)= \frac{1}{2}\)` - Conjectures: `\(\tilde{p}^\beta = \tilde{p}^\sigma = 0\)` otherwise ??? SLOW SLOW SLOW --- Efficient Equilibrium Payoffs ===================================== .center[] --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_U.png) background-size:contain --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_V.png) background-size:contain --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_X.png) background-size:contain --- Constructing Equilibria ==================================== - Efficient equilibrium: everyone invests - Suppose realized prices: `\(p(1,1)= \frac{1}{2}\)` - Conjectures: `\(\tilde{p}^\beta = \tilde{p}^\sigma = 0\)` otherwise - Coordination failure: no one invests - Suppose realized prices: `\(p(0,0)=0\)` - Conjectures: - Buyers: `\(\tilde{p}^\beta (1,1) = 1\)` - Sellers: `\(\tilde{p}^\sigma(1,1) = 0\)` ??? SLOW SLOW SLOW --- Coordination Failure Payoffs ===================================== .center[] -- - To prevent buyer deviating: `\(\tilde{p}^\beta (1,1) \geq \frac{3}{4}\)` - To prevent seller deviating: `\(\tilde{p}^\sigma(1,1) \leq \frac{1}{4}\)` -- - **Result**: Every coordination involves contradictory conjectures - `\(\tilde{p}^\beta (1,1) \neq \tilde{p}^\sigma(1,1)\)` ??? Only optimize, given prices. Think deeper: Need to think price will be 1 and that will clear market --- Refinement ======================= - Multiplicity from a Nash-style equilibrium - But the coordination failure requires contradictory conjectures -- - To discipline conjectures and give predictive power, I introduce a mild refinement - Trembling-hand perfection (Selten 1975) - Simple way to capture mistakes/experimentation - Show which equilibria are robust/stable to small mistakes ??? Robert Lucas taught us to beware of theorists bearing free parameters --- Perturbed Strategies ==================================== When defining trembles, 1. Symmetric trembling hand only at investment stage: - Each investment must be chosen with positive probability `\(\epsilon(b), \epsilon(s) >0\)` by each buyer and seller 1. Law of large numbers: in the aggregate each investment must be chosen by at least `\(\epsilon(b)\)` buyers and `\(\epsilon(s)\)` sellers -- - Definition: A perfect equilibrium is limit of some sequence of equilibria for `\(\epsilon = (\epsilon(b),\epsilon(s)) \to (0,0)\)` - **Claim**: The unique perfect equilibrium *allocation* is efficient ??? Symmetric: all players and all actions have same tremble --- Uniqueness Proof ========== - With trembling hand, each `\(b\)` and `\(s\)` are played `\(\Rightarrow\)` `\(\tilde{p}^\beta(b,s) = \tilde{p}^\sigma(b,s) = p(b,s)\)` - Prices are no longer a free parameter -- - Step 1: Besides `\(p(1,1)\)` fix all other prices at 0 - Verify ex post these prices are part of equilibrium --- Deviating from `\((0,0)\)` ================================================ - Step 2: For `\(\epsilon >> 0\)` .center[] -- - For `\(p(1,1) > \frac{1}{4}\)`, all sellers choose `\((1,1)\)` -- - For `\(p(1,1) < \frac{3}{4}\)`. all buyers choose `\((1,1)\)` -- - `\(\Rightarrow\)` For any price, someone wants to deviate from `\((0,0)\)` - `\((0,0)\)` cannot be voluntarily chosen in an equilibrium --- Supply and Demand for `\((1,1)\)` ========== Step 3: For any `\(\epsilon >> 0\)`, again some price `\(p(1,1) \in \left[\frac{1}{4}, \frac{3}{4}\right]\)` clears market <!--- - At that price, everyone wants to invest if they can match ---> <!--- - `\(\text{min}\left\{1- \epsilon(b), 1- \epsilon(s)\right\}\)` invest on both sides ---> --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_L.png) background-size:contain --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_M.png) background-size:contain --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_N.png) background-size:contain --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/tikz_O.png) background-size:contain --- .center[] - For every `\(\epsilon \to (0,0)\)`, the match `\((1,1)\)` is the unique equilibrium strategy choice `\(\blacksquare\)` -- - However, prices are not unique - *e.g.* if equilibrium payoffs are `\(\frac{1}{4}\)`, other prices can be in `\(\left[0, \frac{1}{4}\right)\)` -- - Indirectly proved no mixed strategy equilibrium (endogenous trembles) - This is different from 2-entrepreneur example <!-- - If someone is mixing between `\((0,0)\)` and `\((1,1)\)`, `\(p(1,1)\)` is public and either buyers or sellers strictly prefer `\((1,1)\)` --> --- Results for the Example ============================== 1. Inefficiencies (coordination failures) can occur under competition 2. However, coordination failures require contradictory conjectures 3. Experimentation eliminates contradictions `\(\Rightarrow\)` no coordination failures 4. [All perfect, competitive equilibria are efficient](#fulltalk) -- - These results hold in a general environment --- Extending the Results ================================ - **Main Theorem**: for a general class of matching models with investments, all *perfect* investment equilibria are efficient -- - General finite types, `\(\beta \in X,\sigma \in Y\)` - General finite investment levels - General cost of investment that can depend on types, `\(c\left(\beta, b\right)\)` and `\(c\left(\sigma, s\right)\)` - General surplus function `\(v(b,s)\)` --- Not *So* Simple ==================================== As in any paper, technical complications arise, *e.g.* I need to 1. Model an explicit competitive matching market, given fixed investments, compared to picking a match ex ante 1. Integrate the non-cooperative investment stage with the competitive matching market 1. Define a tremble and perfect equilibrium with a continuum of players - [Example highlights the core ideas](#conclusions) --- Structure of General Proof ================================ - General buyers' problem: `\(\max_{b,s} v(b,s) - \tilde{p}^\beta (b,s) - c\left(\beta, b\right)\)` -- - General sellers' problem: `\(\max_{b,s} \tilde{p}^\sigma (b,s) - c\left(\sigma, s\right)\)` -- - Total welfare, given measure over types `\(\mathcal{E}(\beta), \mathcal{E}(\sigma)\)` `$$\sum_\beta \left[ \max_{b,s} v(b,s) - \tilde{p}^\beta (b,s)- c\left(\beta, b\right)\right] \mathcal{E}(\beta) + \sum_\sigma \left[\max_{b,s} \tilde{p}^\sigma (b,s) - c\left(\sigma, s\right)\right]\mathcal{E}(\sigma)$$` -- - Efficiency `$$\max_{b,s} \left\{ \sum_\beta \left[v(b,s) - p(b,s)- c\left(\beta, b\right)\right]\mathcal{E}(\beta) + \sum_\sigma\left[p(b,s) - c\left(\sigma, s\right)\right] \mathcal{E}(\sigma)\right\}$$` -- - Since each agent is choosing both `\((b,s)\)`, total welfare = efficiency `\(\blacksquare\)` --- name:fulltalk Road Map for Talk ==================================== 1. Example 2. **General Matching Model** - Competitive Equilibria with Fixed Investments - Conditional Efficiency 3. Investment Game where Players Choose Investments - Investment Equilibria - Perfect Investment Equilibria - Efficiency --- Stage 1: Types ==================================== - General endowed types: `\(t \in T\)` - Partitioned into - Buyer endowed types: `\(\beta \in X\)` - Seller endowed types: `\(\sigma \in Y\)` - Economy: a positive measure on `\(T\)`: `\(\mathcal{E} \in M_+(T)\)` --- Stage 1: Investments ==================================== - General investments: `\(a \in A\)` - Partitioned into - Buyer investments: `\(b \in B\)` - Seller investments: `\(s \in S\)` -- - Type determines cost of acquiring investment `$$c : T \times A \to \mathbb{R} \cup {\infty}$$` - `\(c(t,a)\)` is the cost to a type `\(t\)` of investment `\(a\)` --- Stage 2: Matching Game Distribution ==================================== - Individual choices lead to distribution of investments `\(\mu \in M_+(A)\)` - Not keeping track of `\(M_+(T \times A)\)` - For any investment `\(a\)`, `\(\mu(a)\)` is the mass of individuals with investment `\(a\)` - Buyers: `\(\mu_b \in M_+(B)\)` - Sellers: `\(\mu_s \in M_+(S)\)` --- Stage 2: Matching Game ==================================== - To allow individuals to remain unmatched, let `\(B^{\emptyset} \equiv B \cup \emptyset\)` and `\(S^{\emptyset} \equiv S \cup \emptyset\)` - Bounded value function: `\(v : B^{\emptyset} \times S^{\emptyset} \to \mathbb{R}\)` - Competitive prices: `\(p : B^{\emptyset} \times S^{\emptyset} \to \mathbb{R}\)` - Normalize: `\(p(b,\emptyset) \equiv p(\emptyset,s) \equiv 0\)` -- - Could generalize to `\(v(\beta,b,\sigma,s)\)`, if sufficient prices `\(p(\beta,b,\sigma,s)\)` - To focus on investment choice and coordination failures, I don't allow this --- Stage 2: Matching Game ==================================== - A matching is a measure `$$x \in M_+ (B^{\emptyset} \times S^{\emptyset})$$` - A matching `\(x\)` is **feasible** for `\(\mu\)` if `\(x(\emptyset,\emptyset)=0\)`, and `$$\sum_{s' \in S^{\emptyset}}x(b,s') = \mu(b)~~~\forall b$$` `$$\sum_{b' \in B^{\emptyset}}x(b', s) = \mu(s)~~~\forall s$$` --- Stage 2: Equilibrium ==================================== <font color=#125972><b>Definition:</b></font> The pair `\((x,p)\)` is an (ex post) **competitive equilibrium** for `\(\mu\)` if `\(x\)` is feasible for `\(\mu\)`, -- - For each `\(b \in \text{ supp } \mu_b\)` and each `\((b,s^*) \in \text{ supp } x,\)` the match maximizes `\(b\)`'s utility: -- `$$s^* \in \text{ argmax }_{s \in \text{ supp } \mu_s }\left\{ v(b,s) - p(b,s)\right\},$$` -- - For each `\(s \in \text{ supp } \mu_s\)` and each `\((b^*,s) \in \text{ supp } x\)`, the match maximizes `\(s\)`'s utility: -- `$$b^* \in \text{ argmax }_{s \in \text{ supp } \mu_b } \left\{p(b,s)\right\}.$$` ??? Walk slowly through each --- Stage 2: Efficiency ==================================== - Social matching gains function given `\(\mu\)` `$$g(\mu) \equiv \text{max}_{x} \sum_{b \in B^{\emptyset}} \sum_{s \in S^{\emptyset}} v(b,s) x(b,s)~~~ s.t. x \text{ is feasible given } \mu$$` - An allocation that attains `\(g(\mu)\)` is **conditionally efficient** --- Stage 2: Efficiency ==================================== - **Conditional First Welfare Theorem**: If a pair `\((x,p)\)` is competitive for `\(\mu\)`, then it is conditionally efficient - It is conditional because maximization only holds within the support - `\(\text{supp }\mu\)` may not include the ex ante efficient `\(b\)` and `\(s\)` - Given the choice in a non-cooperative setting, do people choose the efficient `\(b\)` and `\(s\)`? --- Road Map for Talk ==================================== 1. Example 2. General Matching Model - Competitive Equilibria with Fixed Investments - Conditional Efficiency 3. **Investment Game where Players Choose Investments** - Investment Equilibria - Perfect Investment Equilibria - Efficiency --- Stage 1: Investments ==================================== - Fix the ex ante population `\(\mathcal{E}\)` - An allocation of investments is a measure `\(\nu \in M_+(T\times A)\)` - Marginals: `\(\nu_{T}, \nu_A\)` - `\(\mu = \nu_{A}\)` - An allocation `\(\nu\)` is *feasible* for `\(\mathcal{E}\)` if `\(\nu_T = \mathcal{E}\)` - Each agent of type `\(t\)` has price conjectures: `\(\tilde{p}^t : B^{\emptyset} \times S^{\emptyset} \to \mathbb{R}\)` ??? `\(\nu\)` is not v$ --- Conjectured Indirect Utility ==================================== - Define conjectured indirect utility of matching - Buyer with `\(b\)` who conjectures `\(\tilde{p}^\beta\)` `$$v^*_b\left(\tilde{p}^\beta\right) = \max_{s \in S^\emptyset}\left\{ v(b,s) - \tilde{p}^\beta(b,s)\right\}$$` - Seller with `\(s\)` who conjectures `\(\tilde{p}\)` `$$v^*_s\left(\tilde{p}^\sigma\right) = \max_{b \in B^\emptyset} \left\{\tilde{p}^\sigma(b,s)\right\}$$` --- Investment Equilibrium ==================================== - A tuple `\(\left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)\)` is an **investment equilibrium** for `\(\mathcal{E}\)` if `\(\nu\)` is feasible, `\((x,p)\)` is a competitive equilibrium for `\(\nu_A\)`, Buyers optimize their investment: for all `\((t,b) \in \text{supp } \nu\)` `$$v_b^*(\tilde{p}^\beta) - c(t,b) \geq v^*_{b'}(\tilde{p}^\beta) - c(t,b') ~~~\forall b' \in B$$` Sellers optimize: for all `\((s,t) \in \text{supp } \nu\)` `$$v_s^*(\tilde{p}^\sigma) - c(s,t) \geq v^*_{s'}(\tilde{p}^\sigma) - c(s',t) ~~~\forall s' \in S$$` Rational conjectures: $$\tilde{p}^t(b,s) = p(b,s) ~~\forall (b,s) \in \text{ supp } \mu_b \times \text{ supp } \mu_s $$ --- Efficiency ==================================== - Total cost of investments `\(\nu\)` is `\(\sum_A \sum_T c(t,a) \nu(t,a).\)` - Total surplus from `\(\nu\)` is `$$G(\nu) = g(\nu_{A}) - \sum_A \sum_T c(t,a) \nu(t,a).$$` - The allocation `\(\nu\)` is unconditionally **efficient** for `\(\mathcal{E}\)` if it is feasible and `\(G(\nu) \geq G(\nu')\)` for all other feasible allocation `\(\nu'\)` --- Unconstrained conjectures ==================================== - Without more structure, there will often be many equilibria - Suppose not matching is generates no value generation: `\(v(\emptyset,s)=v(b,\emptyset)=0\)` - Often assumed in buyer/seller markets - `\(b=0, s=0\)` can be part of an equilibrium through appropriate conjectures - No matter how large `\(v(b,s)\)` is, sellers can still rationally conjecture that `\(p(b,s)=0\)` -- - Weakness of Nash equilibrium: - In this class of matching games, the surplus maximizing **and minimizing** outcomes can occur --- Need for Refinement ==================================== - Off-path conjectures are a free parameter - Why would sellers conjecture `\(\tilde{p}^t(1,1) = 0\)`? -- - Economists have recognized this issue in other competitive contexts: - Zame (2007) - "imposing no discipline would admit equilibria which are **viable only because different agents hold contradictory conjectures**" -- - Gale (1992) - "Typically, there exists a large number of equilibria and **some refinement of the equilibrium concept is required to give the theory predictive power**. One such refinement is based on the notion of the 'trembling' hand." --- Perturbations ==================================== - Consider a perturbed strategy vector for all buyers, `\(\epsilon_B = (\epsilon(b))_{b \in B},\)` satisfying `\(\epsilon(b)>0\)` for all `\(b \in B\)` and `$$\sum_B \epsilon(b) \leq 1$$` - For all sellers `\(\epsilon_S = (\epsilon(s))_{s \in S},\)` satisfying `\(\epsilon(s)>0\)` for all `\(s \in S\)` and `$$\sum_S \epsilon(s) \leq 1$$` - A perturbed game is indexed by the vector of perturbed strategies `\(\epsilon = (\epsilon_B,\epsilon_S)\)` --- `\(\epsilon\)`-Investment Equilibria ==================================== - An allocation `\(\nu(\epsilon)\)` is `\(\epsilon\)`-feasible for `\(\mathcal{E}\)` if `\(\nu_T = \mathcal{E}\)` and for all `\(a \in A\)` `$$\nu_{A}(\epsilon(a)) \geq \epsilon(a)$$` - A tuple `\(\left(\nu(\epsilon), \left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)\)` is an `\(\epsilon\)`-investment equilibrium for `\(\mathcal{E}\)` if `\(\nu\)` is `\(\epsilon\)`-feasible, `\(p\)` is a competitive price for `\(\nu_A\)`, and for all `\((t,a)\)` such that `\(\nu_{A}(\epsilon) > \epsilon\)`, `$$v_a^*(\tilde{p}^t) - c(t,a) \geq v^*_{a'}(\tilde{p}^t) - c(a',t) ~~~\forall a' \in A$$` -- - Note that by construction, `\(\text{ supp } \nu_{A}(\epsilon) = A\)` -- - A tuple `\(\left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)\)` is a **perfect investment equilibria** if there exists a sequence of `\(\epsilon\)`, such that `\(\lim_{k \to \infty} M(\epsilon^k)=0\)` such that `\(\left(\nu(\epsilon),\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right) \to \left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)\)`. --- Restricted Efficiency ==================================== - The allocation `\(\nu(\epsilon)\)` is `\(\epsilon\)`-efficient for `\(\mathcal{E}\)` if it is feasible and `\(G(\nu(\epsilon)) \geq G(\nu'(\epsilon))\)` for all other `\(\epsilon\)`-feasible allocation `\(\nu'\)` - **Lemma 1**: If `\((\nu(\epsilon), p)\)` is an `\(\epsilon\)`-investment equilibrium, then it is `\(\epsilon\)`-efficient. -- - Proof: Let `\(Q(\epsilon)\)` be the utility generated by the trembling actions `$$Q(\epsilon)= \sum_b \left\{\sum_\beta \left[v^*_b\left(\tilde{p}^\beta\right) - c(\beta,b)\right]\nu_T(\beta)\right\}\epsilon(b)$$` `$$+ \sum_s \left\{\sum_\sigma \left[v^*_s\left(\tilde{p}^\sigma\right) - c(\sigma,s)\right]\nu_T(\sigma)\right\}\epsilon(s).$$` --- Proof (continued) ==================================== - Hence, `$$\sum_\beta \left[ \max_{b,s} v(b,s) - \tilde{p}^\beta (b,s)- c\left(\beta, b\right)\right] \nu_T(\beta)\left( 1 - \epsilon(b)\right)$$` `$$+ \sum_\sigma \left[\max_{b,s} \tilde{p}^\sigma (b,s) - c\left(\sigma, s\right)\right]\nu_T(\sigma)\left( 1 - \epsilon(s)\right)$$` <!--$$\underbrace{\left( \left[\max_b v(b,s) - \tilde{p}^\beta (b,s) - c\left(\beta, b\right)\right] + \left[\max_s \tilde{p}^\sigma (b,s) - c\left(\sigma, s\right)\right] \right) \left(1 - \sum_a \epsilon(a) \right)}_{\text{Optimized Choice}}$$--> $$ + \underbrace{Q(\epsilon)}_{\text{Constrained Choice}}$$ - But since all actions are played by trembles, `\(\tilde{p}^\beta(b,s) =\tilde{p}^\sigma(b,s)\)` - Therefore they optimize the entire left expression - It is exactly the complete market problem, which is efficient `\(\qquad \blacksquare\)` --- Full Efficiency ==================================== - **Theorem**: If `\(\left(\nu,\left\{ \tilde{p}^t\ \right\}_{t\in T},p,x\right)\)` is a perfect investment equilibrium, then it is efficient. -- - Proof: Immediate from Lemma 1, since `\(\underbrace{Q(\epsilon)}_{\text{Constrained Choice}} \to 0\)` `\(\qquad \qquad \qquad \qquad \blacksquare\)` --- Common Conjectures Justifications ==================================== - Predictive power comes from imposing more restrictions on conjectures than just rational conjectures - Trembling hand + large market `\(\Rightarrow\)` as-if complete markets - Complete markets means rational conjectures `\(\Rightarrow\)` common conjectures -- - Other approaches: - Dubey and Geanakoplos (2002): fictitious seller who contributes an infinitesimal to each health insurance pool - Dubey, Geanakoplos, and Shubik (2005): government intervenes to sell infinitesimal quantities of each asset and fully delivers on its promises - Zame (2007): only considers "common-conjecture" equilibria --- name:conclusions Food for Applied Thought ==================================== - "Big push" policies attempts to move economies/sectors from coordination failure to more efficient outcome - Rosenstein-Rodan (1943), Murphy, Shleifer & Vishny (1988) -- - Idea: government policy can force investment, so other actors' best-reply is to also invest - If works, very low cost way to increase output --- <img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_A.png"> --- <img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_D.png"> --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/jde.png) background-size:contain --- background-image: url(https://briancalbrecht.github.io/docs/slides/images/jmp/jep.png) background-size:contain --- <img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_D.png"> --- <img src="https://briancalbrecht.github.io/docs/slides/images/jmp/tikz2_E.png"> --- Conclusion ==================================== - Under competition, coordination failures rely contradictory conjectures - If people can disagree, then of course there are many equilibria - If we want predictive power, we must use a refinement - I use trembling-hand perfection Main theorem: Modified First Welfare Theorem .center[Even with **incomplete and endogenous markets**,] .center[every perfect investment equilibrium is efficient.] --- class: center, middle # Thank You <br> ###
Paper: [bit.ly/Albrecht-JMP](https://bit.ly/Albrecht-JMP) <br> ###
Slides: [bit.ly/Albrecht-JMP-Slides](https://bit.ly/Albrecht-JMP-Slides) <br> <!--###
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